Complex SVG file fails to import correctly into Blender
The importer for SVG is quite basic (I feel like I keep writing this over and over)
Whatever you can do to simplify the file, will help you when import it to blender.
Effects on SVGs, and nested objects in groups (or in groups within a group within a group) don't translate well.

Un-group all of the objects in the SGV file and import.
A crucial difference between 3d graphics and 2D illustration is that in 3D you can't have surfaces occupying the same plane.
On 2D Illustration object's layers just occlude each other, but when imported to to 3D software, they will be created on top of each other causing Z-fighting(when two or more surfaces occupy the same space and the render engine cannot differentiate them). In 2D graphics you can get away with overlapping planes in layers, in 3D you can't. Coplanar faces cannot be on to of each other: no two faces can occupy the same plane.

How to fix this? You would need to edit the paths on the original file, and make sure that no paths are overlapping or intersecting. You will need boolean operations like path union, difference, intersect, exclusion, etc. to clean up.
Or you might choose to separate some of the elements on different files, and import them individually to blender, and edit them there, by joining different curves into the same object.
Since you are dealing with a single basic shape, and multiples copies of it, so you can get rid of most of the file to make your life easier, once you have the basic shapes you can make instances in blender.

Here's a picture of the file after cleaning up in inkscape, mostly using boolean exclusion and difference.

Ideally you should be dealing with only four curves that contain all of the elements with no overlaps: